Descartes' rule of signs

In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting the zero coefficients), and that the difference between these two numbers is always even. This implies in particular that, if this difference is zero or one, then there is exactly zero or one positive root, respectively. In mathematics, Descartes' rule of signs, first described by René Descartes in his work La Géométrie, is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting the zero coefficients), and that the difference between these two numbers is always even. This implies in particular that, if this difference is zero or one, then there is exactly zero or one positive root, respectively. By a homographic transformation of the variable, one may use Descartes' rule of signs for getting a similar information on the number of roots in any interval. This is the basic idea of Budan's theorem and Budan–Fourier theorem. By repeating the division of an interval into two intervals, one gets eventually a list of disjoints intervals containing together all real roots of the polynomial, and containing each exactly one real root. Descartes rule of signs and homographic transformations of the variable are, nowadays, the basis of the fastest algorithms for computer computation of real roots of polynomials (see Real-root isolation). Descartes himself used the transformation x → –x for using his rule for getting information of the number of negative roots. The rule states that if the terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign differences between consecutive nonzero coefficients, or is less than it by an even number. Multiple roots of the same value are counted separately. As a corollary of the rule, the number of negative roots is the number of sign changes after multiplying the coefficients of odd-power terms by −1, or fewer than it by an even number. This procedure is equivalent to substituting the negation of the variable for the variable itself.For example, to find the number of negative roots of f ( x ) = a x 3 + b x 2 + c x + d {displaystyle f(x)=ax^{3}+bx^{2}+cx+d} , we equivalently ask how many positive roots there are for − x {displaystyle -x} in Using Descartes' rule of signs on g ( x ) {displaystyle g(x)} gives the number of positive roots x i {displaystyle x_{i}} of g, and since g ( x ) = f ( − x ) {displaystyle g(x)=f(-x)} it gives the number of positive roots ( − x i ) {displaystyle (-x_{i})} of f, which is the same as the number of negative roots x i {displaystyle x_{i}} of f.